Hello, I have proved this statement using induction, but I am just wondering if you can do it without induction:

for all .

I'm wondering if there is another way since I heard that induction is "not elegant". (Wondering)

Thanks!

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- Jun 25th 2013, 02:42 PMRagnarokProving an inequality without induction
Hello, I have proved this statement using induction, but I am just wondering if you can do it without induction:

for all .

I'm wondering if there is another way since I heard that induction is "not elegant". (Wondering)

Thanks! - Jun 25th 2013, 04:11 PMemakarovRe: Proving an inequality without induction
Induction in general is very elegant. In this case, obviously, it requires strengthening the induction statement from to for some positive expression E(n). This method illustrates the reason for the inequality and shows how close the sum gets to 2. I would say this is very illuminating way of proving the inequality. Which E(n) did you use?

Another way is a standard trick from generating functions. Let . Then for (see Wikipedia), so . - Jun 29th 2013, 06:46 PMSorobanRe: Proving an inequality without induction
Hello, Ragnarok!

Quote:

Hello, I have proved this statement using induction,

but I am just wondering if you can do it without induction:

. . .

We have: . . . . .

Multiply by

Subtract: . . . .

. . The geometric series has a sum of:

We have: .

. . . . . . . .

Multiply by 2: .

. . . . . . . . . . .

. . . . . . . . . . .

Since the fraction is positive,

- Jun 30th 2013, 10:45 AMRagnarokRe: Proving an inequality without induction
Thank you both so much! I'm sorry I didn't get back to you sooner. emakarov, I tried to put my proof in the form you described and I couldn't - in fact, I couldn't even reproduce the proof. So maybe I got something wrong. I'm going to try again. Soroban, I love your answer, thank you so much.

- Jun 30th 2013, 03:04 PMemakarovRe: Proving an inequality without induction
I am not sure my method, which is supposed to work by proving for some E(n), is feasible. However, Soroban's final equality can be proved by induction.

- Jun 30th 2013, 03:05 PMRagnarokRe: Proving an inequality without induction
Okay, I think I got it. Here's what I did for the inductive step:

Since (can be shown separately), we have and so

as required.

So what would be the in this argument? - Jun 30th 2013, 03:11 PMRagnarokRe: Proving an inequality without induction
Oh, sorry, I forgot to put it back in the right form:

I hope that's right... - Jun 30th 2013, 04:17 PMRagnarokRe: Proving an inequality without induction
Mistake in that last bit...should be an in parts of it.